At the Henderson Middle School awards ceremony, the principal is going to honor outstanding students with gift cards. The gift cards for excellent grades, g, are worth $40, and the gift cards for perfect attendance, a, are worth $25. The principal has $4,000 allotted for this event, and he is going to recognize 100 students. Write the pair of linear equations that model this situation.
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Let x be the number of excellent grades gift cards and y be the number of perfect attendance gift cards.
Since the school has allotted $4,000 for this event, the total value of the gift cards cannot exceed $4,000.
40x + 25y ≤ 4,000
The principal is going to recognize 100 students, so the total number of gift cards given out must be 100.
x + y = 100
Therefore, the two linear equations that model this situation are:
40x + 25y ≤ 4,000
x + y = 100
Oh, I see what we're doing here! Let me put on my mathematician's hat.
Let's use g to represent the number of gift cards for excellent grades and a to represent the number of gift cards for perfect attendance.
For the first equation, we know that each excellent grade gift card is worth $40. Since we want the total value of excellent grade gift cards to be $4,000, our equation is:
40g = 4000
For the second equation, we know that each perfect attendance gift card is worth $25. Similarly, we want the total value of perfect attendance gift cards to be $4,000, so our equation is:
25a = 4000
So, our pair of linear equations is:
40g = 4000
25a = 4000
Now, let's see if we can figure out the values of g and a!
To write the pair of linear equations that model this situation, we need to define the variables and set up equations based on the given information.
Let's use:
- "g" to represent the number of excellent grades gift cards.
- "a" to represent the number of perfect attendance gift cards.
Based on the information provided, we know that:
1. The gift cards for excellent grades (g) are worth $40 each.
2. The gift cards for perfect attendance (a) are worth $25 each.
3. The total number of students to be recognized is 100.
4. The total amount allotted for this event is $4,000.
Given these facts, we can now set up the equations:
Equation 1: The value of the excellent grades gift cards must equal the total amount allotted.
The value of each excellent grades gift card is $40, and there are "g" excellent grades gift cards. Therefore:
40g = total value of excellent grades gift cards
Equation 2: The value of the perfect attendance gift cards must also equal the total amount allotted.
The value of each perfect attendance gift card is $25, and there are "a" perfect attendance gift cards. Therefore:
25a = total value of perfect attendance gift cards
Additionally, we know that the total number of students to be recognized is 100:
g + a = 100
Therefore, the pair of linear equations that model this situation are:
Equation 1: 40g = total value of excellent grades gift cards
Equation 2: 25a = total value of perfect attendance gift cards
Equation 3: g + a = 100
Let's define two variables to represent the number of gift cards given for excellent grades (g) and perfect attendance (a), respectively.
Let x represent the number of gift cards for excellent grades (g).
Let y represent the number of gift cards for perfect attendance (a).
According to the given information, the gift cards for excellent grades are worth $40, so the total value for excellent grades is 40x. Similarly, the gift cards for perfect attendance are worth $25, so the total value for perfect attendance is 25y.
The principal has $4,000 allotted for this event, so the total value of all the gift cards should be equal to $4,000. Therefore, we can write the first equation as:
40x + 25y = 4000
The principal is going to recognize a total of 100 students. So, the number of gift cards for excellent grades and perfect attendance should add up to 100. Hence, the second equation is:
x + y = 100
Therefore, the pair of linear equations that model this situation are:
40x + 25y = 4000
x + y = 100